Central limit theorems describing isolation by distance under various forms of power-law dispersal

TL;DR

This paper investigates new asymptotic patterns of genetic isolation by distance under various power-law dispersal models, extending previous stochastic PDE approaches to include non-local coalescence phenomena.

Contribution

It generalizes existing models by allowing different regions for parent sampling and offspring dispersal with power-law distributions, revealing new coalescence behaviors.

Findings

Local and non-local coalescence are possible under power-law dispersal.

The model incorporates different sampling and dispersal regions, broadening previous frameworks.

New asymptotic patterns of isolation by distance are characterized.

Abstract

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $\Lambda$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and V\'eber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians.